Auxiliary gradient-based sampling algorithms

We introduce a new family of Markov chain Monte Carlo samplers that combine auxiliary variables, Gibbs sampling and Taylor expansions of the target density. Our approach permits the marginalization over the auxiliary variables, yielding marginal samplers, or the augmentation of the auxiliary variables, yielding auxiliary samplers. The well-known Metropolis-adjusted Langevin algorithm MALA and preconditioned Crank-Nicolson-Langevin algorithm pCNL are shown to be special cases. We prove that marginal samplers are superior in terms of asymptotic variance and demonstrate cases where they are slower in computing time compared with auxiliary samplers. In the context of latent Gaussian models we propose new auxiliary and marginal samplers whose implementation requires a single tuning parameter, which can be found automatically during the transient phase. Extensive experimentation shows that the increase in efficiency (measured as the effective sample size per unit of computing time) relative to (optimized implementations of) pCNL, elliptical slice sampling and MALA ranges from tenfold in binary classification problems to 25 fold in log-Gaussian Cox processes to 100 fold in Gaussian process regression, and it is on a par with Riemann manifold Hamiltonian Monte Carlo sampling in an example where that algorithm has the same complexity as the aforementioned algorithms. We explain this remarkable improvement in terms of the way that alternative samplers try to approximate the eigenvalues of the target. We introduce a novel Markov chain Monte Carlo sampling scheme for hyperparameter learning that builds on the auxiliary samplers. The MATLAB code for reproducing the experiments in the paper is publicly available and an on-line supplement to this paper contains additional experiments and implementation details.